We define and develop the infrastructure of homotopical inverse diagrams in categories with attributes (CwA's). Specifically, given a category with attributes C and an ordered homotopical inverse category I, we construct the category with attributes C^I of homotopical diagrams of shape I in C and Reedy types over these, and we show how various logical structure (Pi-types, identity types, and so on) lifts from the original CwA to the diagram CwA. This may be seen as providing a general class of diagram models of type theory, and forms a companion paper to arXiv:1610.00037, "The homotopy theory of type theories" , which applies the present results in constructing semi-model structures on categories of contextual categories.

中文翻译：

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具有属性的类别中的同伦逆图

我们在具有属性的类别（CwA）中定义和开发同伦逆图的基础结构。具体来说，给定一个具有属性 C 的类别和一个有序的同伦逆类别 I，我们构造了具有 C 中形状 I 的同伦图的属性 C^I 和 Reedy 类型的类别，并展示了各种逻辑结构（Pi 类型） 、身份类型等）从原始 CwA 提升到图 CwA。这可以看作是提供了类型理论的一般图模型类，并形成了 arXiv:1610.00037 的配套论文，“类型理论的同伦理论”，该论文将当前结果应用于构建上下文类别的半模型结构。类别。